Origins of the phrasal verb "to fall asleep".

The particular meaning of "fall" is fossilized in several phrases, like "to fall in love," "to fall ill" - it's not about the feeling of falling asleep, but rather an archaic usage of "becoming (a state)."

• To fall in love is attested from 1520s; to fall asleep is late 14c. To fall down is early 13c. (a-dun follon); to fall behind is from 1856. Fall through "fail, come to nothing" is from 1781. To fall for something is from 1903.

(Etymonline)

• a 1300 Cursor M. 7428 Þe king he sal gar fall o-slepe.-

The use of the phrasal verb is attested in Middle English from 1393:

1393 Langland Piers Plowman C. xxii. 5 Ich fel eft-sones a slepe.

[From "fall, v.". OED Online. September 2015. Oxford University Press. http://www.oed.com/view/Entry/67829?rskey=76QoIB&result=2&isAdvanced=false (accessed October 15, 2015).]

This use draws on a verbal sense of 'fall' attested possibly around 1225:

38. a. Of persons: To pass (usually, with suddenness) †in, into, †to, upon some specified condition, bodily or mental, or some external condition or relation.

?c1225 (▸?a1200) Ancrene Riwle (Cleo. C.vi) (1972) 166 He..swa feol into unhope.

(op. cit.)